There is a connected subset of $𝐑^2$ which is not path-connected.

hbghlyj

A2-metricspaces.pdf

Theorem 7.4.3. There is a connected subset of $𝐑^2$ which is not path-connected.

Proof. There is a classic example, known as the Topologist's sine-curve. This is the set $A \subseteq \mathbf{R}^{2}$ given by
\[
\{(0, y):-1 \leqslant y \leqslant 1\} \cup\{(x, \sin (1 / x): x \in(0,1]\} .
\]
Why is $A$ connected? It is quite easy to convince oneself that $A=\bar{E}$, where $E=\{(x, \sin (1 / x): x \in(0,1]\}$. However, $E$ is connected, being the image of the connected set $(0,1]$ under a continuous map, and so the connectedness of $A$ is immediate from Lemma 7.1.5.

Why is $A$ not path-connected? It is "intuitively clear" that there is no path $\gamma:[0,1] \rightarrow A$ with $\gamma(0)=(0,0)$ and $\gamma(1)=(1, \sin (1))$, but we must prove this. Suppose we have such a path $\gamma$. Write $\ell$ for the vertical line $\{0\} \times[-1,1]$, thus $A=E \cup \ell$. Since $\ell$ is closed in $A, \gamma^{-1}(\ell)$ is closed, and in particular contains its supremum $t$. Thus $\gamma(t) \in \ell$, whilst $\gamma(u) \in E$ for all $u>t$.

Let $p_{Y}: \mathbf{R}^{2} \rightarrow \mathbf{R}$ be projection onto the $y$-coordinate, i.e. $p_{Y}(x, y)=y$. Since $p_{Y}$ is continuous, so is the composition $p_{Y} \circ \gamma:[0,1] \rightarrow \mathbf{R}$. Thus there is some $\delta>0$ such that
(7.2) $\quad\left|p_{Y}\left(\gamma\left(u_{1}\right)\right)-p_{Y}\left(\gamma\left(u_{2}\right)\right)\right| \leqslant 1$ for all $u_{1}, u_{2} \in[t, t+\delta]$

Now let $p_{X}$ be projection onto the $x$-coordinate, i.e. $p_{X}(x, y)=x$. The composition $p_{X} \circ \gamma$ is continuous, and so by the intermediate value theorem and the fact that $p_{X}(\gamma(t+\delta))>0,\left(p_{X} \circ \gamma\right)[t, t+\delta]$ contains some interval $[0, c], c>0$.

However, as $x$ ranges over $(0, c], \sin (1 / x)$ takes all values in $[-1,1]$ (infinitely often), so there are $u_{1}, u_{2} \in[t, t+\delta]$ such that $p_{Y}\left(\gamma\left(u_{1}\right)\right)=1, p_{Y}\left(\gamma\left(u_{2}\right)\right)=-1$. This contradicts (7.2).

hbghlyj

An Asymptote tutorial

Here's one last example that uses parametric graphing to plot the function f(x) = sin(1/x), the “topologist's sine curve,” which is notoriously difficult to.